Average Error: 0.7 → 0.6
Time: 3.9s
Precision: 64
\[\frac{e^{a}}{e^{a} + e^{b}}\]
↓
\[e^{a - \log \left(e^{a} + e^{b}\right)}\]
\frac{e^{a}}{e^{a} + e^{b}}
↓
e^{a - \log \left(e^{a} + e^{b}\right)}
double f(double a, double b) {
        double r157530 = a;
        double r157531 = exp(r157530);
        double r157532 = b;
        double r157533 = exp(r157532);
        double r157534 = r157531 + r157533;
        double r157535 = r157531 / r157534;
        return r157535;
}
↓
double f(double a, double b) {
        double r157536 = a;
        double r157537 = exp(r157536);
        double r157538 = b;
        double r157539 = exp(r157538);
        double r157540 = r157537 + r157539;
        double r157541 = log(r157540);
        double r157542 = r157536 - r157541;
        double r157543 = exp(r157542);
        return r157543;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.7
Target0.0
Herbie0.6
\[\frac{1}{1 + e^{b - a}}\]

Derivation

  1. Initial program 0.7

    \[\frac{e^{a}}{e^{a} + e^{b}}\]
  2. Using strategy rm

  3. Applied add-exp-log0.7

    \[\leadsto \frac{e^{a}}{\color{blue}{e^{\log \left(e^{a} + e^{b}\right)}}}\]

  4. Applied div-exp0.6

    \[\leadsto \color{blue}{e^{a - \log \left(e^{a} + e^{b}\right)}}\]

  5. Final simplification0.6

    \[\leadsto e^{a - \log \left(e^{a} + e^{b}\right)}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1 (+ 1 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))